Abstract

Some mechanical models are proposed in this paper in order to understand the decomposition into irreducible representations of some unitary reducible representations of the Lorentz group ${\rm SO}(3,1)$. The action of ${\rm SO}(3,1)$ on the space-time is not transitive, so the following three orbits are considered: $Q\sb 1=\{x\sp µ\vert x\sp µx\sb µ=1, x\sp 0\geq 1\}$, $Q\sb 0=\{x\sp µ\vert x\sp µx\sb µ=0, x\sp 0\geq 0\}$, $Q\sb {-1}=\{x\sp µ\vert x\sp µx\sb µ=-1\}$. Let $\scr H\sb \epsilon$ $(\epsilon=1,0,-1)$ be the Hilbert space of square integrable functions on the orbit $Q\sb \epsilon$. A unitary representation (UR) $\scr U\sb \epsilon$ on $\scr H\sb \epsilon$ is defined by $(\scr U\sb \epsilon(\Lambda)\psi)(x)=\psi(\Lambda\sp {-1}x)$, $\Lambda\in{\rm SO}(3,1)$. This UR is not irreducible and it decomposes into a direct sum or integral of UIRs of ${\rm SO}(3,1)$ running only over the principal series. Remember that the principal series can be labelled by two parameters $(j\sb 0,\rho)$ and they are $j\sb 0=0$, $\rho\in\bold R\sp {\geq 0}$, $\scr C\sb 1=-1-\rho\sp 2$, $\scr C\sb 2=0$; $j\sb 0=\frac 12,1,\frac 32,\cdots$, $\rho\in\bold R$, $\scr C\sb 1=-1-\rho\sp 2+j\sp 2\sb 0$, $\scr C\sb 2=-j\sb 0\rho$, where $\scr C\sb 1$ and $\scr C\sb 2$ are the two ${\rm SO}(3,1)$ Casimir operators.
The representations $\scr U\sb 1$ and $\scr U\sb 0$ are direct integrals of the UIR $(j\sb 0=0, \rho\in\bold R\sp {\geq 0})$; however a direct sum of the UIRs $(j\sb 0=\frac 12, 1,\frac 32,\cdots$, $\rho=0)$ also appears in the decomposition of ${\scr U}\sb {-1}$. The mechanical models have as configuration spaces the spaces $Q\sb \epsilon$ and as phase spaces their associated cotangent bundles $T\sp *Q\sb \epsilon$. Let us consider the cases for $\epsilon=1$ and $-1$. Starting from the Lagrangian $\scr L=-\frac 12\dot q{}\sp µ\dot q\sb µ$ the Hamiltonian obtained after the Legendre transformation is $H=-\frac 12b\sp µb\sb µ=-\epsilon\scr C\sb 1/2$, $b\sp µ=-\dot q{}\sp µq\sp µq\sb µ=\epsilon$ and $q\sp µb\sb µ=0$. Note that the ${\rm SO}(3,1)$ generators can be written as $S\sb {µ\nu}=-q\sp µb\sb \nu+q\sp \nu b\sb µ$. Integrating the equations of motion in phase space, $\dot q{}\sp µ=\{q\sp µ,H\}=-b\sp µ$, $\dot b{}\sp µ=\{b\sp µ,H\}=-\epsilon b\sp 2q\sp µ$, we see that their projection in the configuration space $Q\sb \epsilon$ gives unbounded motion for $Q\sb 1$ and the Casimir operator is negative. For $Q\sb {-1}$, bounded and unbounded motions appear according as to whether $b\sp µ$ is a spacelike, timelike or lightlike quadrivector, respectively. For bounded motion the Casimir operator is positive, and negative in the other case. Finally, for the case of $Q\sb 0$ with Hamiltonian $H=-\frac 12\scr C\sb 1=\frac 12(q\sp ip\sp i)\sp 2\geq 0$ $(i=1,2,3)$ the motion is also unbounded and the Casimir operator negative.
In conclusion, the different signs of the Casimir operator give rise to different behaviours of these mechanical systems in agreement with the decomposition pattern of their associated UR of ${\rm SO}(3,1)$.